On a Method for Solving Non-Stationary Heat Conduction Problems with Constant over Time Internal Heat Sources

摘要

The mathematical description of the heat transfer in bodies with internal heat sources processes poses difficulties. To obtain solutions of boundary-value problems describing these processes, exact (Fourier method, integral transformations, etc.), approximate (collocation, Galerkin method) analytical methods are used, as well as numerical methods (finite differences, finite elements, etc.). The solutions obtained by using exact analytical methods are expressed by complex functional dependencies and are not suitable for engineering. The most significant difficulties are problems with nonlinear sources of heat, periodic action, etc. The use of classical analytical methods to solve them is extremely difficult, and in some cases impossible. Despite all these circumstances, analytical solutions have a number of serious advantages over numerical ones, since they allow performing parametric analysis of the studied processes. Based on the integral heat balance method and additional boundary conditions (characteristics) use, a numerical - analytical solution of the heat conduction problem for an infinite plate under symmetric first kind border conditions with constant power internal sources is obtained. The physical meaning of the boundary conditions is the fulfillment of the initial differential equation at the boundary points of the system under consideration, i.e. the points where the first kind boundary condition is specified. By introducing into consideration a new unknown function, heat flux on the plate surface, a simple form analytical solution of the problem was obtained. It is shown that with an increase in the number of approximations, the residual of the equation being solved decreases, which indirectly indicates the convergence of the method. It is also noted that the approach proposed can be used to solve partial differential equations that exclude the separation of variables.